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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
Yesterday at 11:16 PM
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
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[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
Yesterday at 11:16 PM
0 replies
k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
Israeli Mathematical Olympiad 1995
YanYau   24
N 5 minutes ago by bjump
Source: Israeli Mathematical Olympiad 1995
Let $PQ$ be the diameter of semicircle $H$. Circle $O$ is internally tangent to $H$ and tangent to $PQ$ at $C$. Let $A$ be a point on $H$ and $B$ a point on $PQ$ such that $AB\perp PQ$ and is tangent to $O$. Prove that $AC$ bisects $\angle PAB$
24 replies
YanYau
Apr 8, 2016
bjump
5 minutes ago
P(x), integer, integer roots, P(0) =-1,P(3) = 128
parmenides51   3
N 12 minutes ago by Rohit-2006
Source: Nordic Mathematical Contest 1989 #1
Find a polynomial $P$ of lowest possible degree such that
(a) $P$ has integer coefficients,
(b) all roots of $P$ are integers,
(c) $P(0) = -1$,
(d) $P(3) = 128$.
3 replies
parmenides51
Oct 5, 2017
Rohit-2006
12 minutes ago
2017 CGMO P1
smy2012   9
N 16 minutes ago by Bardia7003
Source: 2017 CGMO P1
(1) Find all positive integer $n$ such that for any odd integer $a$, we have $4\mid a^n-1$
(2) Find all positive integer $n$ such that for any odd integer $a$, we have $2^{2017}\mid a^n-1$
9 replies
smy2012
Aug 13, 2017
Bardia7003
16 minutes ago
Euler's function
luutrongphuc   1
N 32 minutes ago by luutrongphuc
Find all real numbers \(\alpha\) such that for every positive real \(c\), there exists an integer \(n>1\) satisfying
\[
\frac{\varphi(n!)}{n^\alpha\,(n-1)!} \;>\; c.
\]
1 reply
luutrongphuc
an hour ago
luutrongphuc
32 minutes ago
Square problem
Jackson0423   1
N an hour ago by maromex
Construct a square such that the distances from an interior point to the vertices (in clockwise order) are
1,2,3,4, respectively.
1 reply
Jackson0423
an hour ago
maromex
an hour ago
Sequence with infinite primes which we see again and again and again
Assassino9931   4
N an hour ago by SimplisticFormulas
Source: Balkan MO Shortlist 2024 N6
Let $c$ be a positive integer. Prove that there are infinitely many primes, each of which divides at least one term of the sequence $a_1 = c$, $a_{n+1} = a_n^3 + c$.
4 replies
2 viewing
Assassino9931
Apr 27, 2025
SimplisticFormulas
an hour ago
Fermat points of Pentagon
Jackson0423   1
N an hour ago by Jackson0423
It is known that, in general, a pentagon has three Fermat points. But I'm curious—if there are exactly two Fermat points inside the pentagon, under what conditions does the distance sum reach a minimum? Can you help me?
1 reply
Jackson0423
an hour ago
Jackson0423
an hour ago
Inequality , Exponent problem
biit   5
N an hour ago by Jackson0423
If $\ P=(\frac {6375}{6374})^ {6374} $ , $\ Q=(\frac {6375}{6374})^ {6375} $ then prove that $P^{Q}$ >$ Q^{P}$
5 replies
biit
an hour ago
Jackson0423
an hour ago
Ant walks
monishrules   0
an hour ago
Source: Homemade
3 Ants in a plane are placed on the vertices of a equilateral triangle of side length s, each ant moves s unit in a random direction with uniform probability. find the expected change in the area of the equilateral triangle.

some interesting extensions which I expect to only be solve-able via integration.
a) a right angled triangle with legs of side length A, and step length A?
b) a general n-gon?
c) a non uniform probability distribution? given by f(theta)
d) expected increase in volume/surface area of a cube?
0 replies
monishrules
an hour ago
0 replies
Inequality for 4 variables
Nguyenhuyen_AG   0
an hour ago
Let $a, \, b, \, c, \, d$ are non-negative real numbers. Prove that
\[( {a}^{2}-bc ) \sqrt {a+b+c}+ ( {b}^{2}-cd) \sqrt {b+c+d}+ ( {c}^{2}-da ) \sqrt {c+d+a}+ ( {d}^{2}-ab ) \sqrt {d+a+b} \geqslant 0.\]
0 replies
Nguyenhuyen_AG
an hour ago
0 replies
Two equal angles
jayme   1
N an hour ago by jayme
Dear Mathlinkers,

1. ABCD a square
2. I the midpoint of AB
3. 1 the circle center at A passing through B
4. Q the point of intersection of 1 with the segment IC
5. X the foot of the perpendicular to BC from Q
6. Y the point of intersection of 1 with the segment AX
7. M the point of intersection of CY and AB.

Prove : <ACI = <IYM.

Sincerely
Jean-Louis
1 reply
jayme
Today at 6:52 AM
jayme
an hour ago
Symetric inequality
Nguyenhuyen_AG   0
2 hours ago
Let $a, \, b, \, c$ are non-negative real numbers and $k \geqslant 0.$
(i) Prove that
\[\frac{a(a^2-bc)}{\sqrt{ka+b+c}} + \frac{b(b^2-ca)}{\sqrt{kb+c+a}} + \frac{c(c^2-ab)}{\sqrt{kc+a+b}} \geqslant 0.\](ii) Prove that
\[(a^2-bc)\sqrt{ka+b+c}+(b^2-ca)\sqrt{kb+c+a}+(c^2-ab)\sqrt{kc+a+b} \geqslant 0.\]
0 replies
Nguyenhuyen_AG
2 hours ago
0 replies
Lord Evan the Reflector
whatshisbucket   23
N 2 hours ago by bjump
Source: ELMO 2018 #3, 2018 ELMO SL G3
Let $A$ be a point in the plane, and $\ell$ a line not passing through $A$. Evan does not have a straightedge, but instead has a special compass which has the ability to draw a circle through three distinct noncollinear points. (The center of the circle is not marked in this process.) Additionally, Evan can mark the intersections between two objects drawn, and can mark an arbitrary point on a given object or on the plane.

(i) Can Evan construct* the reflection of $A$ over $\ell$?

(ii) Can Evan construct the foot of the altitude from $A$ to $\ell$?

*To construct a point, Evan must have an algorithm which marks the point in finitely many steps.

Proposed by Zack Chroman
23 replies
whatshisbucket
Jun 28, 2018
bjump
2 hours ago
4 variables with quadrilateral sides 2
mihaig   6
N 2 hours ago by mihaig
Source: Own
Let $a,b,c,d\geq0$ satisfying
$$\frac1{a+1}+\frac1{b+1}+\frac1{c+1}+\frac1{d+1}=2.$$Prove
$$\left(a+b+c+d-2\right)^2+8\geq3\left(abc+abd+acd+bcd\right).$$
6 replies
mihaig
Apr 29, 2025
mihaig
2 hours ago
intersections of the angle bisector with intouch triangle
danepale   7
N Sep 24, 2020 by Com10atorics
Source: Middle European Mathematical Olympiad T-5
Let $ABC$ be a triangle with $AB < AC$. Its incircle with centre $I$ touches the sides $BC, CA,$ and $AB$ in the points $D, E,$ and $F$ respectively. The angle bisector $AI$ intersects the lines $DE$ and $DF$ in the points $X$ and $Y$ respectively. Let $Z$ be the foot of the altitude through $A$ with respect to $BC$.

Prove that $D$ is the incentre of the triangle $XYZ$.
7 replies
danepale
Sep 21, 2014
Com10atorics
Sep 24, 2020
intersections of the angle bisector with intouch triangle
G H J
Source: Middle European Mathematical Olympiad T-5
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danepale
99 posts
#1 • 2 Y
Y by Adventure10, Mango247
Let $ABC$ be a triangle with $AB < AC$. Its incircle with centre $I$ touches the sides $BC, CA,$ and $AB$ in the points $D, E,$ and $F$ respectively. The angle bisector $AI$ intersects the lines $DE$ and $DF$ in the points $X$ and $Y$ respectively. Let $Z$ be the foot of the altitude through $A$ with respect to $BC$.

Prove that $D$ is the incentre of the triangle $XYZ$.
Z K Y
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jayme
9789 posts
#2 • 1 Y
Y by Adventure10
Dear Mathlinkers,
an outline of my proof...

1. M the midpoint of BC
2. the triangle MXY is M-isoceles; ZM is the Z-inner bissector of the triangle ZXY
3. M is the center of the circumcircle of DXY (Morel's circle) (see http://jl.ayme.pagesperso-orange.fr/ vol. 4 Symetrique de (OI) p. 5)
4. According to the A-Mention circle (Shamrock theorem), D is the incenter of ZXY.

Sincerely
Jean-Louis
Z K Y
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Vo Duc Dien
341 posts
#3 • 2 Y
Y by Adventure10, Mango247
See attachment.
Attachments:
Z K Y
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hajimbrak
209 posts
#4 • 1 Y
Y by Adventure10
We have $\angle YAF=A/2$ and $\angle YFA=90^0+(C/2)$.Hence $\angle DYX=\angle AYF=(B/2)$.Also $\angle IBD=(B/2)$
Hence $IBDY$ is cyclic.Now $ID\perp BD \implies IY\perp BY$ since $IBDY$ is cyclic.Also $AZ\perp BZ$,hence $ABZY$ is cyclic.
Hence $\angle YZC=\angle BAY=(A/2)$.Similarly we can prove that $\angle XZC=(A/2)$.Hence $DZ$ bisects $\angle XZY$.
Now $\angle BDY=90^0+(C/2)$.Hence $\angle ZYD=180^0-(90^0+(C/2))-(A/2)=(B/2)=\angle DYX$.Hence $DY$ bisects $\angle ZYX$.Hence $D$ is the incentre of $\triangle XYZ$.
Z K Y
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Mikasa
56 posts
#5 • 3 Y
Y by PatrikP, Adventure10, Mango247
Let $P\in AC$ such that $AP=AB$. Let $BP\cap AI=X_0$. Let $EX_0$ meet $\odot DEF$ at $D_0\neq E$. Clearly $D_0$ is on the other side of $BP$ than $F$. Now, $AI$ is the perpendicular bisector of $EF$ and $BP$. Again, $\angle FD_0X_0=\angle ED_0F=\angle AFE=\angle ABP=\angle FBX_0$ i.e. $D_0BFX_0$ is cyclic. Thus, $\angle BDF=\angle BX_0F=90^{\circ}-\angle FX_0A=\angle X_0FE=\angle X_0EF=\angle DEF$. So, $BD_0$ is tangent to $\odot DEF$. This means that $D\equiv D_0$. Thus $X\equiv X_0$. So we proved that $BX\perp AI$.

Now, $\angle DYE=\angle FYE=180^{\circ}-\angle YFE-\angle YEF$. But $Y$ is on the perpendicular bisector of $EF$. So, $\angle DYE=180^{\circ}-2\angle YFE=180^{\circ}-2\angle DFE=180^{\circ}-2\angle DEC=\angle DCE$. Thus $DECY$ is cyclic, and so $\angle IYE=\dfrac{1}{2}\angle DYE=\dfrac{1}{2}\angle DCE=\angle ICE$ which means $IYCE$ is cyclic too. Thus $\angle IYC=\angle IEC=90^{\circ}$. So $CY\perp AI$ too.

Now, $AXZB$ and $AZYC$ both are cyclic. So, $\angle ZXY=90^{\circ}-\angle BXZ=90^{\circ}-\angle BAZ=\angle ABC$ and $\angle DXY=\angle AXE=90^{\circ}-\angle DEF=90^{\circ}-\angle BDF=\angle IBD=\dfrac{1}{2}\angle ABC$. This means $DX$ bisects $\angle ZXY$. Also, $\angle XZD=180^{\circ}-\angle XZB=\angle XAB=\angle IAB=\angle IAC=\angle YAC=\angle YZC$$=\angle YZD$. Thus $DZ$ bisects $\angle XZY$.

All these results imply that $D$ is the incenter of $\triangle XYZ$.
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YaMohammad
17 posts
#6 • 3 Y
Y by godofgeometry, mahditorogh, Adventure10
Hi
Here's my solution.
1)By angle chasing we understand that DXY=B/2 and DYX=C/2 and ZDY=90+B/2
2)If we prove that ZXD=B/2 the problem will be solved.
3)So we need to say that ZXY=B.
4)Because IBD=B/2 so IBDX is cyclic. So XBD=XID but because DI and AZ are parallel we have XID=XAZ so ABZX is cyclic.
5)So ZXY=B so we are done.
:wink:
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dothef1
110 posts
#7 • 1 Y
Y by Adventure10
EDIIT...
This post has been edited 1 time. Last edited by dothef1, Jan 17, 2016, 10:15 AM
Reason: false
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Com10atorics
174 posts
#8 • 2 Y
Y by Mango247, Mango247
$\angle IXD=180-\tfrac {\angle A}{2} -\angle AED=90-\tfrac {\angle A}{2}-\tfrac {\angle B}{2}=\tfrac {\angle C}{2}$
So $IDXE$ is cyclic.
$\angle IXC=180-\angle IFC=90=\angle AZC$
So $AZXC$ is cyclic as well. And now,
$\angle DXZ=\angle EDB-\angle CZX=180-\angle AEX-\tfrac{\angle A}{2}=\angle DXY$
Similarly we show $YD$ bisects $\angle XYZ$ and so $I$ is the incenter $XYZ$
Z K Y
N Quick Reply
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